7.1 Magnetic Fields, Flux & Solenoid Properties

Magnetic Fields, Flux & Solenoid Properties

In electrotechnical engineering, magnetism and electromagnetism form the physical foundation for the operation of electrical motors, generators, transformers, contactors, protective circuit breakers, and measuring instruments. Without magnetic interactions, large-scale generation and distribution of alternating current (AC) electrical energy would be impossible.

Magnetism is an invisible force of attraction or repulsion exerted between magnetic materials or moving electric charges. Materials exhibiting magnetic properties are broadly classed as permanent magnets (which retain their magnetic alignment indefinitely without external excitation) or temporary electromagnets (which exhibit magnetic fields only when excited by an electric current).

---

Principles of Magnetism and Magnetic Field Lines

Every magnet possesses two distinct regions of concentrated magnetic influence known as poles: a North-seeking pole (N) and a South-seeking pole (S). Magnetic monopoles do not exist in nature; if a bar magnet is divided in half, each fragment instantly forms its own pair of North and South poles.

The Fundamental Law of Magnetic Poles

The interaction between magnetic poles follows a universal electrotechnical rule:

• Like poles repel one another (North repels North; South repels South).
• Unlike poles attract one another (North attracts South).

  [ N ]  ---- Repulsion ---->  [ N ]          [ S ]  <---- Attraction ---->  [ N ]

Properties of Magnetic Field Lines (Lines of Magnetic Flux)

The region surrounding a magnet where its magnetic influence can be detected is termed a magnetic field. This field is mapped conceptually using lines of magnetic force, formally designated as lines of magnetic flux. These imaginary lines possess specific physical characteristics:

1. Continuous Closed Loops: Lines of flux have no beginning and no end; they form continuous, unbroken closed paths. Externally, they emerge from the North pole and enter the South pole. Internally through the magnetic material, they complete the closed loop by flowing from South back to North.
2. Never Intersect or Cross: Magnetic field lines can never cross, bend sharply, or touch one another. At any given point in space, the magnetic field has a single unique vector direction.
3. Longitudinal Tension: Lines of flux act like stretched elastic rubber bands under tension along their length, constantly striving to shorten themselves to the minimum possible path length between opposite poles.
4. Lateral Repulsion: Adjacent lines of flux traveling in the same direction exert a mutual sideways (lateral) repulsive force against one another, causing them to spread out evenly across available space.
5. Density Represents Field Strength: The closer together the lines of flux are packed, the stronger the magnetic field in that region. Flux lines are most densely concentrated at the pole faces, where magnetic field intensity is at its peak.

---

Magnetic Quantities and SI Units

To perform engineering calculations on electrical equipment, two fundamental magnetic quantities are defined: Magnetic Flux and Magnetic Flux Density.

1. Magnetic Flux ($\Phi$)

Magnetic flux (symbolized by the Greek letter phi, $\Phi$) represents the total quantity of magnetic field lines produced by a magnetic source or passing through a defined space. It is analogous to electric current in an electrical circuit.

• SI Unit: Weber (symbol Wb), named in honor of German physicist Wilhelm Eduard Weber.
• In base physical concepts, one Weber ($1\text{ Wb}$) corresponds to $10^8$ (100 million) individual magnetic flux lines.

2. Magnetic Flux Density ($B$)

Magnetic flux density (symbol $B$) measures the concentration of magnetic flux per unit area perpendicular to the direction of the field lines. It quantifies how densely packed the magnetic lines are within a given cross-section.

• SI Unit: Tesla (symbol T), named after Serbian-American inventor Nikola Tesla.
• A flux density of one Tesla ($1\text{ T}$) exists when a magnetic flux of one Weber passes perpendicularly through a surface area of one square metre ($1\text{ Wb/m}^2$).

The Flux Density Equation

$B = \frac{\Phi}{A}$

Where:

• $B$ = Magnetic flux density in Teslas ($\text{T}$ or $\text{Wb/m}^2$)
• $\Phi$ = Total magnetic flux in Webers ($\text{Wb}$)
• $A$ = Cross-sectional area perpendicular to the magnetic field in square metres ($\text{m}^2$)

Formula Transpositions

Depending on which variable is required in electrotechnical calculations, the formula transposes as follows:

$\Phi = B \times A \qquad \text{and} \qquad A = \frac{\Phi}{B}$

Step-by-Step Worked Calculations

Worked Example 1: Calculating Flux Density ($B$)

Problem: A transformer core has a rectangular cross-section measuring $50\text{ mm}$ by $40\text{ mm}$. If a total magnetic flux of $0.0036\text{ Wb}$ passes through the core, calculate the magnetic flux density.

Solution:

1. Convert core dimensions to metres and calculate cross-sectional area: $A = 0.050\text{ m} \times 0.040\text{ m} = 0.0020\text{ m}^2 = 2.0 \times 10^{-3}\text{ m}^2$
2. Apply the flux density formula: $B = \frac{\Phi}{A} = \frac{0.0036\text{ Wb}}{0.0020\text{ m}^2} = 1.8\text{ T}$

Worked Example 2: Calculating Total Flux ($\Phi$)

Problem: The cylindrical pole face of an electromagnetic contactor has a diameter of $30\text{ mm}$. If the magnetic flux density across the air gap is uniform at $1.4\text{ T}$, determine the total magnetic flux established.

Solution:

1. Calculate the circular area from radius ($r = 15\text{ mm} = 0.015\text{ m}$): $A = \pi r^2 = \pi \times (0.015\text{ m})^2 \approx 7.0686 \times 10^{-4}\text{ m}^2$
2. Transpose the equation to find flux $\Phi$: $\Phi = B \times A = 1.4\text{ T} \times 7.0686 \times 10^{-4}\text{ m}^2 \approx 9.896 \times 10^{-4}\text{ Wb} = 0.9896\text{ mWb}$

Worked Example 3: Calculating Required Pole Area ($A$)

Problem: An electromagnet must establish a total flux of $6.0\text{ mWb}$ ($0.006\text{ Wb}$) while operating at a maximum permissible core flux density of $1.5\text{ T}$ to prevent magnetic saturation. Calculate the minimum required core cross-sectional area in square centimetres ($\text{cm}^2$).

Solution:

1. Transpose the formula for area $A$: $A = \frac{\Phi}{B} = \frac{0.006\text{ Wb}}{1.5\text{ T}} = 0.004\text{ m}^2$
2. Convert square metres to square centimetres ($1\text{ m}^2 = 10,000\text{ cm}^2$): $A = 0.004 \times 10,000 = 40\text{ cm}^2$

---

Electromagnetism Around a Straight Conductor

In 1820, Danish physicist Hans Christian Oersted discovered that an electric current flowing through a metallic conductor generates an associated magnetic field in the surrounding space.

Magnetic Field Geometry

When a direct current passes through a straight, isolated conductor:

• The magnetic field lines form concentric circles centered on the longitudinal axis of the conductor.
• The field is continuous along the entire length of the conductor.
• The field is strongest immediately adjacent to the conductor surface and weakens with radial distance.

The Right-Hand Grip Rule for Straight Conductors

To determine the direction of the concentric magnetic field lines surrounding a straight conductor, apply the Right-Hand Grip Rule:

1. Imagine grasping the conductor with the right hand.
2. Extend the thumb so it points in the direction of conventional current flow (from positive $+$ to negative $-$).
3. The curled fingers naturally encircle the conductor in the exact direction of the circular magnetic field lines.

               Current (Thumb Points UP)
                          ^
                          |
                 .--------|--------.
                /         |         \
               |    (Curled Fingers) |
                \   Rotate Anticlockwise
                 '--------|--------'
                          |

Drawing Conventions: Dots and Crosses

On two-dimensional electrical engineering schematics and cross-sectional drawings, three-dimensional current directions are denoted using arrow symbolism:

• A Circle with a Cross ($\times$ or $\bigotimes$): Represents current flowing away from the viewer into the plane of the page. This mimics looking at the fletching (feathers) of an arrow receding into the distance. By the Right-Hand Grip Rule, this generates a clockwise magnetic field.
• A Circle with a Dot ($\cdot$ or $\bigodot$): Represents current flowing towards the viewer out of the plane of the page. This mimics looking at the sharp point of an approaching arrow. By the Right-Hand Grip Rule, this generates a counter-clockwise (anticlockwise) magnetic field.

Interaction Between Parallel Conductors

When two parallel conductors carry current simultaneously, their circular magnetic fields interact:

• Currents Flowing in the Same Direction: The opposing flux lines between the conductors cancel out, while the outer lines enclose both conductors, forcing them together (attraction).
• Currents Flowing in Opposite Directions: The flux lines between the conductors travel in the same direction and reinforce one another, crowding together and forcing the conductors apart (repulsion).

---

Solenoids: Helical Coils and Magnetic Field Strengthening

A single straight conductor produces a relatively weak, dispersed magnetic field. When the conductor is wound into a tightly packed, helical coil of many turns, it is called a solenoid.

Solenoid Field Pattern

Inside a solenoid, the circular magnetic fields produced by each individual turn combine and reinforce one another along the coil axis. This creates a dense, uniform, parallel magnetic field running straight through the center of the coil. Externally, the flux lines loop back from one end to the other, creating an overall magnetic field pattern that is virtually identical to that of a permanent bar magnet, complete with distinct North and South pole ends.

Determining Solenoid Magnetic Polarity

Two primary rules are used to identify which end of a solenoid forms the North pole and which forms the South pole:

1. The Right-Hand Grip Rule for Coils:
   • Grasp the solenoid coil with the right hand.
   • Wrap the curled fingers around the coil in the direction of conventional current flow through the windings.
   • The extended thumb points directly toward the North magnetic pole.
2. The End Rule:
   • View the face of the coil directly along its longitudinal axis:
   • If current flows around the face in a clockwise direction, that face is a South pole (visualized by placing arrowheads on the curved tips of the capital letter S).
   • If current flows around the face in a counter-clockwise (anticlockwise) direction, that face is a North pole (visualized by placing arrowheads on the tips of the capital letter N).

      Counter-Clockwise = NORTH                      Clockwise = SOUTH
               ^                                             |
           /-------\                                     /-------\
          |    N    |                                   |    S    |
           \-------/                                     \-------/
                   |                                             ^

Factors Governing Electromagnet Field Strength

The magnetic strength of a solenoid depends on four primary physical parameters:

1. Current Magnitude ($I$): Magnetic flux density is directly proportional to current. Doubling the current doubles the magnetic field strength.
2. Number of Turns ($N$): Each turn adds its own magnetic contribution. Doubling the number of turns doubles the magnetizing influence.
   • The product of current and turn count is termed the Magnetomotive Force (mmf), designated $F_m$: $F_m = I \times N$ Expressed in ampere-turns (A-t), mmf represents the magnetic driving potential that establishes flux through a magnetic circuit, directly analogous to electromotive force (EMF) in an electrical circuit.
3. Length of the Coil ($l$): For a given number of turns, a shorter, more tightly wound coil concentrates the magnetomotive force over a smaller distance, increasing magnetic field intensity ($H = \frac{I N}{l}$).
4. Core Material Permeability ($\mu$):
   • An air-cored solenoid has high magnetic opposition (reluctance), yielding a relatively weak field.
   • Inserting a ferromagnetic core (such as soft iron or silicon steel) provides an effortless path for magnetic lines. The atomic domains in the iron align with the field, multiplying the resulting flux density by a factor of hundreds or thousands ($B = \mu H$).

---

Practical Electrotechnical Applications

Electromagnets are ubiquitous across commercial and industrial electrical installations:

• Contactor and Relay Coils: Energizing a low-current control coil creates an electromagnetic pull that overcomes a return spring, attracting an iron armature to close or open heavy-current main power contacts feeding three-phase motors or heating banks.
• Solenoid Actuators: Provide direct linear mechanical movement to open fluid valves in central heating circuits, release security door latches, or operate water valves in commercial appliances.
• Electric Bells and Buzzers: Utilize a spring-loaded armature that breaks its own electrical circuit via a make-and-break contact, causing rapid cyclic reciprocation and continuous striking of a gong.
• Circuit Breaker Electromagnetic Trips: In Miniature Circuit Breakers (BS EN 60898), excessive short-circuit fault currents flow through an internal electromagnetic solenoid coil, instantly creating sufficient magnetic pull to unlatch the mechanical trip mechanism within 10 to 40 milliseconds, providing instantaneous ADS protection.